Section A: Introduction to Game Theory
- Game theory is primarily concerned with:
a) Decision-making in competitive situations involving two or more participants
b) Inventory replenishment decisions only
c) Project scheduling under certainty
d) Transportation-cost allocation
Answer: a) Decision-making in competitive situations involving two or more participants
- A participant who makes decisions in a game is called a:
a) Payoff
b) Player
c) Strategy value
d) Saddle point
Answer: b) Player
- A complete plan of action available to a player is called a:
a) Payoff
b) Game value
c) Strategy
d) Decision tree
Answer: c) Strategy
- The numerical outcome associated with a pair of strategies is called the:
a) Dominance value
b) Probability
c) Criterion value
d) Payoff
Answer: d) Payoff
- A matrix showing the outcomes for different strategy combinations is called a:
a) Payoff matrix
b) Transportation table
c) Assignment matrix only
d) Decision network
Answer: a) Payoff matrix
- In a two-person game, the number of players is:
a) One
b) Two
c) Three
d) Unlimited
Answer: b) Two
- In a zero-sum game, one player’s gain is:
a) Independent of the other player’s loss
b) Greater than the total payoff
c) Exactly equal to the other player’s loss
d) Always zero
Answer: c) Exactly equal to the other player’s loss
- The algebraic sum of payoffs to all players in a zero-sum game is:
a) Positive
b) Negative
c) Variable
d) Zero
Answer: d) Zero
- In a two-person zero-sum game, the payoff matrix is usually written from the viewpoint of:
a) One selected player
b) Both players simultaneously
c) A third party
d) The government
Answer: a) One selected player
- The row player is commonly referred to as:
a) Player B
b) Player A
c) The referee
d) The neutral player
Answer: b) Player A
- The column player is commonly referred to as:
a) Player A
b) The maximizing player
c) Player B
d) The passive player
Answer: c) Player B
- In the usual payoff convention, Player A attempts to:
a) Minimize the number of strategies
b) Avoid all positive payoffs
c) Maximize Player B’s payoff
d) Maximize the payoff shown in the matrix
Answer: d) Maximize the payoff shown in the matrix
- In the usual payoff convention, Player B attempts to:
a) Minimize Player A’s payoff
b) Maximize Player A’s payoff
c) Eliminate every strategy
d) Make all payoffs equal to zero
Answer: a) Minimize Player A’s payoff
- A game is called finite when:
a) Every payoff is zero
b) Each player has a finite number of strategies
c) There is only one player
d) The game ends after one second
Answer: b) Each player has a finite number of strategies
- A pure strategy means that a player:
a) Randomly selects among all strategies
b) Uses no strategy
c) Chooses one strategy with probability one
d) Changes strategies after every move
Answer: c) Chooses one strategy with probability one
- A mixed strategy means that a player:
a) Always uses the same strategy
b) Selects only dominated strategies
c) Chooses the strategy with the largest label
d) Uses two or more strategies according to specified probabilities
Answer: d) Uses two or more strategies according to specified probabilities
- A strategy selected with probability one is:
a) A pure strategy
b) A mixed strategy
c) A dominated strategy only
d) An infeasible strategy
Answer: a) A pure strategy
- The probabilities assigned to a player’s mixed strategies must:
a) Be negative
b) Sum to one
c) Sum to zero
d) All be equal
Answer: b) Sum to one
- In a mixed strategy, every probability must be:
a) Greater than one
b) Negative
c) Between zero and one inclusive
d) An integer greater than zero
Answer: c) Between zero and one inclusive
- A game in which players cooperate to achieve a joint outcome is called:
a) A zero-sum game only
b) A dominance game
c) A saddle-point game
d) A cooperative game
Answer: d) A cooperative game
- A game in which players act independently and do not form binding agreements is:
a) A noncooperative game
b) A transportation game
c) A deterministic queue
d) A replacement game
Answer: a) A noncooperative game
- The value of a game represents:
a) The number of strategies
b) The expected payoff under optimal play
c) The largest entry in the matrix
d) The smallest entry in the matrix
Answer: b) The expected payoff under optimal play
- A positive game value generally favors:
a) Player B
b) Neither player
c) Player A
d) The player with fewer strategies
Answer: c) Player A
- A negative game value generally favors:
a) Player A
b) Both players equally
c) The player with more strategies
d) Player B
Answer: d) Player B
- A game value of zero is commonly described as:
a) A fair game
b) An impossible game
c) A cooperative game
d) A dominated game
Answer: a) A fair game
Section B: Maximin and Minimax Principles
- The maximin principle is applied by:
a) Player B
b) Player A
c) Both players only after dominance
d) A neutral observer
Answer: b) Player A
- Under the maximin principle, Player A first identifies:
a) The maximum entry in each row
b) The minimum entry in each column
c) The minimum payoff in each row
d) The average payoff in each row
Answer: c) The minimum payoff in each row
- After finding each row minimum, Player A chooses the:
a) Smallest row minimum
b) Largest matrix entry
c) Smallest column maximum
d) Largest row minimum
Answer: d) Largest row minimum
- The largest of the row minima is called the:
a) Maximin value
b) Minimax value
c) Expected value
d) Dominance value
Answer: a) Maximin value
- The minimax principle is applied by:
a) Player A
b) Player B
c) The row player only
d) Neither player
Answer: b) Player B
- Under the minimax principle, Player B first identifies:
a) The minimum entry in each row
b) The average of every column
c) The maximum payoff in each column
d) The smallest matrix entry
Answer: c) The maximum payoff in each column
- After finding each column maximum, Player B selects the:
a) Largest column maximum
b) Largest row minimum
c) Average column maximum
d) Smallest column maximum
Answer: d) Smallest column maximum
- The smallest of the column maxima is called the:
a) Minimax value
b) Maximin value
c) Fair value
d) Dominance value
Answer: a) Minimax value
- A saddle point exists when:
a) Every payoff is positive
b) The maximin value equals the minimax value
c) There are only two strategies
d) The game value is zero
Answer: b) The maximin value equals the minimax value
- If the maximin value is less than the minimax value, the game:
a) Has several saddle points automatically
b) Has no strategies
c) Has no saddle point
d) Is always unfair
Answer: c) Has no saddle point
- When a saddle point exists, optimal strategies are:
a) Always mixed
b) Always dominated
c) Unnecessary
d) Pure strategies
Answer: d) Pure strategies
- A saddle-point entry is simultaneously:
a) The minimum in its row and maximum in its column
b) The maximum in its row and minimum in its column
c) The largest entry in the matrix
d) The smallest entry in the matrix
Answer: a) The minimum in its row and maximum in its column
- The game value in a saddle-point game equals:
a) The sum of all payoffs
b) The saddle-point payoff
c) The average of row minima
d) The difference between maximin and minimax
Answer: b) The saddle-point payoff
- If the row minima are 2, 5 and 1, the maximin value is:
a) 1
b) 2
c) 5
d) 8
Answer: c) 5
- If the column maxima are 7, 4 and 9, the minimax value is:
a) 9
b) 7
c) 20
d) 4
Answer: d) 4
- If the maximin value and minimax value are both 6, then:
a) A saddle point exists with game value 6
b) The game has no solution
c) The game value is zero
d) Mixed strategies are compulsory
Answer: a) A saddle point exists with game value 6
- If maximin equals 3 and minimax equals 8, then:
a) The saddle point is 5.5
b) No saddle point exists
c) Player A must use the first row
d) The value is 8
Answer: b) No saddle point exists
- Player A’s maximin rule reflects a:
a) Maximax approach
b) Risk-seeking approach only
c) Best choice under the worst possible response
d) Random strategy approach
Answer: c) Best choice under the worst possible response
- Player B’s minimax rule seeks to:
a) Maximize Player A’s minimum gain
b) Select the largest payoff
c) Ignore Player A’s actions
d) Minimize the maximum possible loss
Answer: d) Minimize the maximum possible loss
- Which inequality always holds in a finite two-person zero-sum game?
a) Maximin value is less than or equal to minimax value
b) Maximin value is always greater than minimax value
c) Maximin equals zero
d) Minimax is always negative
Answer: a) Maximin value is less than or equal to minimax value
- If maximin exceeds minimax, this usually indicates:
a) A mixed-strategy solution
b) A calculation or interpretation error
c) Several saddle points
d) A fair game
Answer: b) A calculation or interpretation error
- The row selected by the maximin principle is the row having:
a) The largest individual payoff
b) The lowest average payoff
c) The greatest row minimum
d) The smallest row maximum
Answer: c) The greatest row minimum
- The column selected by the minimax principle is the column having:
a) The largest column minimum
b) The largest average
c) The smallest entry
d) The smallest column maximum
Answer: d) The smallest column maximum
- If a payoff is the smallest in its row and largest in its column, it is a:
a) Saddle point
b) Dominated payoff
c) Mixed-strategy probability
d) Linear-programming variable
Answer: a) Saddle point
- In a saddle-point game, neither player benefits by:
a) Calculating row minima
b) Unilaterally changing the optimal pure strategy
c) Identifying the game value
d) Using the payoff matrix
Answer: b) Unilaterally changing the optimal pure strategy
Section C: Saddle Points and Pure Strategies
- Consider the matrix (\begin{bmatrix}3&2\4&1\end{bmatrix}). The row minima are:
a) 3 and 4
b) 2 and 4
c) 2 and 1
d) 3 and 1
Answer: c) 2 and 1
- For the matrix (\begin{bmatrix}3&2\4&1\end{bmatrix}), the maximin value is:
a) 4
b) 3
c) 1
d) 2
Answer: d) 2
- For the matrix (\begin{bmatrix}3&2\4&1\end{bmatrix}), the column maxima are:
a) 4 and 2
b) 3 and 1
c) 4 and 1
d) 3 and 2
Answer: a) 4 and 2
- For the matrix (\begin{bmatrix}3&2\4&1\end{bmatrix}), the minimax value is:
a) 4
b) 2
c) 3
d) 1
Answer: b) 2
- The matrix (\begin{bmatrix}3&2\4&1\end{bmatrix}) has a saddle point at:
a) Row 1, Column 1
b) Row 2, Column 1
c) Row 1, Column 2
d) Row 2, Column 2
Answer: c) Row 1, Column 2
- The value of the game (\begin{bmatrix}3&2\4&1\end{bmatrix}) is:
a) 1
b) 3
c) 4
d) 2
Answer: d) 2
- In a pure-strategy solution, each player assigns probability:
a) One to the selected strategy
b) One-half to every strategy
c) Zero to every strategy
d) More than one to the selected strategy
Answer: a) One to the selected strategy
- If a game has multiple saddle points with the same payoff, then:
a) No solution exists
b) More than one optimal pure-strategy pair may exist
c) The game value is undefined
d) Mixed strategies are prohibited
Answer: b) More than one optimal pure-strategy pair may exist
- A saddle point may be found by comparing:
a) Row averages and column averages
b) Largest matrix and smallest matrix entries
c) Maximin and minimax values
d) Only positive and negative entries
Answer: c) Maximin and minimax values
- If every entry in a payoff matrix is increased by 5, the game value:
a) Remains unchanged
b) Is multiplied by 5
c) Becomes zero
d) Increases by 5
Answer: d) Increases by 5
- Adding the same constant to every payoff generally:
a) Does not change the optimal strategies
b) Changes every optimal strategy
c) Eliminates dominance
d) Creates a saddle point in every game
Answer: a) Does not change the optimal strategies
- Multiplying every payoff by a positive constant generally:
a) Reverses the players’ objectives
b) Preserves optimal strategies and scales the game value
c) Makes the game fair
d) Eliminates all saddle points
Answer: b) Preserves optimal strategies and scales the game value
- Multiplying every payoff by a negative constant will:
a) Preserve the roles of players without change
b) Leave the value unchanged
c) Reverse preferences and require careful reinterpretation
d) Always create a zero-sum game
Answer: c) Reverse preferences and require careful reinterpretation
- A pure-strategy equilibrium in a two-person zero-sum game corresponds to:
a) A dominated row
b) A random choice
c) A linear-programming slack variable
d) A saddle point
Answer: d) A saddle point
- If Player A’s guaranteed payoff is 7 and Player B can hold A to 7, then:
a) The game value is 7
b) The game value is zero
c) No equilibrium exists
d) The payoff must be negative
Answer: a) The game value is 7
- A matrix with equal maximin and minimax values is said to be:
a) Strictly dominated
b) Strictly determined
c) Unbounded
d) Infeasible
Answer: b) Strictly determined
- A strictly determined game can be solved by:
a) Linear programming only
b) Graphical methods only
c) Pure-strategy analysis
d) Simulation only
Answer: c) Pure-strategy analysis
- Which condition is unnecessary when a saddle point exists?
a) Calculating row minima
b) Calculating column maxima
c) Determining the game value
d) Solving for mixed-strategy probabilities
Answer: d) Solving for mixed-strategy probabilities
- If the saddle-point payoff is negative, the game favors:
a) Player B
b) Player A
c) Both players equally
d) Neither player
Answer: a) Player B
- If the saddle-point payoff is positive, the game favors:
a) Player B
b) Player A
c) Neither player
d) The player with fewer strategies
Answer: b) Player A
- A zero saddle-point payoff implies:
a) Player A always loses
b) Player B always loses
c) The game is fair under optimal play
d) The game has no solution
Answer: c) The game is fair under optimal play
- In a payoff matrix written for Player A, a large positive entry is:
a) Favorable to Player B
b) Irrelevant
c) Always a saddle point
d) Favorable to Player A
Answer: d) Favorable to Player A
- In a payoff matrix written for Player A, a large negative entry is generally:
a) Favorable to Player B
b) Favorable to Player A
c) A guaranteed saddle point
d) A probability
Answer: a) Favorable to Player B
- The pure strategy selected by Player A at a saddle point is the row containing:
a) The largest row maximum
b) The maximin payoff
c) The smallest matrix entry
d) The highest average payoff only
Answer: b) The maximin payoff
- The pure strategy selected by Player B at a saddle point is the column containing:
a) The smallest column minimum
b) The largest matrix entry
c) The minimax payoff
d) The lowest average only
Answer: c) The minimax payoff
Section D: Dominance Rule
- The dominance principle is used mainly to:
a) Increase the number of strategies
b) Change the game value
c) Convert a game into a cooperative game
d) Eliminate inferior strategies
Answer: d) Eliminate inferior strategies
- For Player A, one row dominates another if it has:
a) Payoffs greater than or equal to the other row in every column
b) Smaller payoffs in every column
c) The same average only
d) More negative entries
Answer: a) Payoffs greater than or equal to the other row in every column
- Since Player A maximizes payoff, a dominated row is generally the row with:
a) Larger entries throughout
b) Smaller or equal entries throughout
c) More columns
d) A larger average only
Answer: b) Smaller or equal entries throughout
- For Player B, one column dominates another if it has:
a) Larger entries in every row
b) The same total only
c) Smaller or equal entries in every row
d) More positive entries
Answer: c) Smaller or equal entries in every row
- Since Player B minimizes Player A’s payoff, a dominated column generally has:
a) Smaller entries throughout
b) Fewer entries
c) A lower average only
d) Larger or equal entries throughout
Answer: d) Larger or equal entries throughout
- If Row 1 is ((5,7,6)) and Row 2 is ((3,4,2)), then:
a) Row 1 dominates Row 2
b) Row 2 dominates Row 1
c) Neither row dominates
d) Both rows dominate each other
Answer: a) Row 1 dominates Row 2
- If Column 1 is ((2,4,3)) and Column 2 is ((5,7,6)), then for Player B:
a) Column 2 dominates Column 1
b) Column 1 dominates Column 2
c) Neither column dominates
d) Both columns are identical
Answer: b) Column 1 dominates Column 2
- Eliminating a strictly dominated strategy:
a) Changes the optimal game value
b) Guarantees a saddle point
c) Does not affect the optimal solution
d) Makes probabilities invalid
Answer: c) Does not affect the optimal solution
- The dominance method is especially useful for:
a) Increasing matrix dimensions
b) Calculating expected value directly
c) Converting payoffs to probabilities
d) Reducing the size of a payoff matrix
Answer: d) Reducing the size of a payoff matrix
- Dominance may occur between:
a) Rows or columns
b) Only rows
c) Only columns
d) Only diagonal entries
Answer: a) Rows or columns
- A strategy may be dominated by:
a) Only one pure strategy
b) Another pure strategy or a combination of strategies
c) The game value only
d) A probability greater than one
Answer: b) Another pure strategy or a combination of strategies
- Dominance by a convex combination means a strategy is inferior to:
a) The largest matrix entry
b) A single saddle point
c) A weighted combination of other strategies
d) The average game value only
Answer: c) A weighted combination of other strategies
- When checking row dominance for Player A, the preferred row has:
a) Lower entries
b) A smaller total only
c) More negative values
d) Higher or equal entries across all columns
Answer: d) Higher or equal entries across all columns
- When checking column dominance for Player B, the preferred column has:
a) Lower or equal entries across all rows
b) Higher entries across all rows
c) The largest total
d) More positive values
Answer: a) Lower or equal entries across all rows
- If two rows are identical, one of them may be:
a) Converted into a column
b) Removed without changing the game
c) Assigned probability two
d) Treated as a saddle point automatically
Answer: b) Removed without changing the game
- If two columns are identical, one column may be:
a) Multiplied by zero
b) Assigned a negative probability
c) Eliminated without changing the game
d) Converted to a row
Answer: c) Eliminated without changing the game
- Dominance should be applied:
a) Only once
b) Only after mixed-strategy calculations
c) Only to square matrices
d) Repeatedly until no further strategies can be eliminated
Answer: d) Repeatedly until no further strategies can be eliminated
- If Row A is ((4,6)) and Row B is ((4,5)), then:
a) Row A weakly dominates Row B
b) Row B dominates Row A
c) Neither row dominates
d) Both rows must remain
Answer: a) Row A weakly dominates Row B
- If Column X is ((3,2)) and Column Y is ((3,5)), then:
a) Column Y dominates Column X
b) Column X weakly dominates Column Y
c) Neither column dominates
d) Both columns are pure strategies
Answer: b) Column X weakly dominates Column Y
- Strict dominance requires the dominating strategy to be:
a) Equal in every outcome
b) Better in only one outcome
c) Better in every relevant outcome
d) Selected with probability one
Answer: c) Better in every relevant outcome
- Weak dominance means a strategy is:
a) Worse in every outcome
b) Equal only in average payoff
c) Always optimal
d) At least as good in all outcomes and better in at least one
Answer: d) At least as good in all outcomes and better in at least one
- Which row should Player A eliminate if Row 1 has higher payoffs than Row 2 in every column?
a) Row 2
b) Row 1
c) Both rows
d) Neither row
Answer: a) Row 2
- Which column should Player B eliminate if Column 1 has lower payoffs than Column 2 in every row?
a) Column 1
b) Column 2
c) Both columns
d) Neither column
Answer: b) Column 2
- A reduced matrix obtained through valid dominance has:
a) A different game value
b) No optimal strategies
c) The same game value as the original matrix
d) Only positive entries
Answer: c) The same game value as the original matrix
- The main benefit of dominance before solving mixed strategies is:
a) It makes every game fair
b) It guarantees equal probabilities
c) It creates negative payoffs
d) It simplifies the calculations
Answer: d) It simplifies the calculations
Section E: Mixed Strategies in 2 × 2 Games
- Mixed strategies are generally required when:
a) No saddle point exists
b) A saddle point exists
c) Every payoff is equal
d) Only one strategy is available
Answer: a) No saddle point exists
- In a 2 × 2 game, Player A usually assigns probabilities:
a) (p) and (p)
b) (p) and (1-p)
c) (p) and (1+p)
d) (p) and (-p)
Answer: b) (p) and (1-p)
- In a 2 × 2 game, Player B usually assigns probabilities:
a) (q) and (q)
b) (q) and (1+q)
c) (q) and (1-q)
d) (q) and (-q)
Answer: c) (q) and (1-q)
- Under optimal mixed strategies, Player A chooses probabilities that make Player B:
a) Prefer the first column only
b) Prefer the second column only
c) Avoid all strategies
d) Indifferent between the columns used
Answer: d) Indifferent between the columns used
- Under optimal mixed strategies, Player B chooses probabilities that make Player A:
a) Indifferent between the rows used
b) Prefer the first row only
c) Prefer the second row only
d) Avoid all strategies
Answer: a) Indifferent between the rows used
- For the payoff matrix (\begin{bmatrix}a&b\c&d\end{bmatrix}), the denominator used in 2 × 2 formulas is:
a) (a+b+c+d)
b) (a-b-c+d)
c) (a+b-c-d)
d) (ad-bc)
Answer: b) (a-b-c+d)
- For (\begin{bmatrix}a&b\c&d\end{bmatrix}), the probability that Player A uses Row 1 is:
a) ((a-c)/(a-b-c+d))
b) ((a-b)/(a-b-c+d))
c) ((d-c)/(a-b-c+d))
d) ((d-b)/(a-b-c+d))
Answer: c) ((d-c)/(a-b-c+d))
- For (\begin{bmatrix}a&b\c&d\end{bmatrix}), the probability that Player A uses Row 2 is:
a) ((d-c)/(a-b-c+d))
b) ((a-c)/(a-b-c+d))
c) ((a-d)/(a-b-c+d))
d) ((a-b)/(a-b-c+d))
Answer: d) ((a-b)/(a-b-c+d))
- For (\begin{bmatrix}a&b\c&d\end{bmatrix}), the probability that Player B uses Column 1 is:
a) ((d-b)/(a-b-c+d))
b) ((d-c)/(a-b-c+d))
c) ((a-b)/(a-b-c+d))
d) ((a-c)/(a-b-c+d))
Answer: a) ((d-b)/(a-b-c+d))
- For (\begin{bmatrix}a&b\c&d\end{bmatrix}), the probability that Player B uses Column 2 is:
a) ((d-b)/(a-b-c+d))
b) ((a-c)/(a-b-c+d))
c) ((a-b)/(a-b-c+d))
d) ((d-c)/(a-b-c+d))
Answer: b) ((a-c)/(a-b-c+d))
- The value of a 2 × 2 zero-sum game without a saddle point is:
a) ((a+b+c+d)/(a-b-c+d))
b) ((a-d)/(b-c))
c) ((ad-bc)/(a-b-c+d))
d) ((a+d)/(b+c))
Answer: c) ((ad-bc)/(a-b-c+d))
- The expected payoff under optimal mixed strategies equals:
a) The largest payoff
b) The smallest payoff
c) Zero in every game
d) The value of the game
Answer: d) The value of the game
- If an optimal probability equals zero, the corresponding strategy:
a) Is not used in the optimal mix
b) Must be used every time
c) Has the highest payoff
d) Is necessarily a saddle point
Answer: a) Is not used in the optimal mix
- If an optimal probability equals one, the corresponding strategy is:
a) Dominated
b) Used as a pure strategy
c) Used half the time
d) Infeasible
Answer: b) Used as a pure strategy
- In an optimal mixed strategy, probabilities are selected to:
a) Maximize the number of strategies
b) Make all payoffs positive
c) Protect a player against exploitation by the opponent
d) Eliminate the game value
Answer: c) Protect a player against exploitation by the opponent
- Randomization is useful because it:
a) Guarantees the largest payoff every time
b) Removes uncertainty
c) Makes all strategies identical
d) Prevents the opponent from predicting the next action with certainty
Answer: d) Prevents the opponent from predicting the next action with certainty
- Consider the matrix (\begin{bmatrix}4&0\0&2\end{bmatrix}). The denominator is:
a) 6
b) 4
c) 2
d) 8
Answer: a) 6
- For (\begin{bmatrix}4&0\0&2\end{bmatrix}), the probability that Player A uses Row 1 is:
a) (1/2)
b) (1/3)
c) (2/3)
d) (1/4)
Answer: b) (1/3)
- For (\begin{bmatrix}4&0\0&2\end{bmatrix}), the probability that Player A uses Row 2 is:
a) (1/3)
b) (1/2)
c) (2/3)
d) (3/4)
Answer: c) (2/3)
- For (\begin{bmatrix}4&0\0&2\end{bmatrix}), the value of the game is:
a) 2
b) 1
c) (4/3)
d) (8/6=4/3)
Answer: d) (8/6=4/3)
- For (\begin{bmatrix}4&0\0&2\end{bmatrix}), Player B uses Column 1 with probability:
a) (1/3)
b) (1/2)
c) (2/3)
d) (3/4)
Answer: a) (1/3)
- For (\begin{bmatrix}4&0\0&2\end{bmatrix}), Player B uses Column 2 with probability:
a) (1/3)
b) (2/3)
c) (1/4)
d) (1/2)
Answer: b) (2/3)
- In a mixed-strategy equilibrium, each player’s expected payoff from every strategy used with positive probability is:
a) Different
b) Always zero
c) Equal to the game value
d) Equal to the largest matrix entry
Answer: c) Equal to the game value
- A strategy that yields less than the game value against the opponent’s optimal mix will generally receive:
a) Probability one
b) Equal probability
c) A negative probability
d) Zero probability
Answer: d) Zero probability
- The method of oddments is used mainly to solve:
a) 2 × 2 games
b) Transportation problems
c) Queuing models
d) Assignment problems
Answer: a) 2 × 2 games
Section F: Graphical and Algebraic Solution Methods
- The graphical method is commonly used for games of size:
a) 3 × 3 only
b) 2 × n or m × 2
c) m × n with any dimensions
d) 1 × 1 only
Answer: b) 2 × n or m × 2
- A 2 × n game has:
a) Two columns and n rows
b) Two players and n payoffs
c) Two rows and n columns
d) Two saddle points
Answer: c) Two rows and n columns
- An m × 2 game has:
a) Two rows and m columns
b) m players and two strategies
c) One row and two columns
d) m rows and two columns
Answer: d) m rows and two columns
- In the graphical method for a 2 × n game, the horizontal axis usually represents:
a) The probability assigned to one of Player A’s two strategies
b) The game value only
c) Player B’s payoff
d) The number of columns
Answer: a) The probability assigned to one of Player A’s two strategies
- In a 2 × n graphical solution, each column of Player B generates:
a) A probability table only
b) A straight payoff line
c) A saddle point automatically
d) A nonlinear constraint
Answer: b) A straight payoff line
- For Player A in a 2 × n game, the relevant boundary is generally the:
a) Upper envelope
b) Average envelope
c) Lower envelope of the payoff lines
d) Vertical axis only
Answer: c) Lower envelope of the payoff lines
- Player A chooses the point on the lower envelope that:
a) Minimizes payoff
b) Has zero probability
c) Lies nearest the origin
d) Maximizes the guaranteed payoff
Answer: d) Maximizes the guaranteed payoff
- In an m × 2 game analyzed from Player B’s perspective, the relevant boundary is generally the:
a) Upper envelope
b) Lower envelope
c) Horizontal axis
d) Average of all lines
Answer: a) Upper envelope
- Player B chooses the point on the upper envelope that:
a) Maximizes Player A’s payoff
b) Minimizes the maximum payoff
c) Makes every line parallel
d) Gives probability zero to both columns
Answer: b) Minimizes the maximum payoff
- The intersection of two relevant payoff lines often identifies:
a) A dominated strategy only
b) The largest matrix entry
c) The optimal mixed-strategy probability and game value
d) The total number of strategies
Answer: c) The optimal mixed-strategy probability and game value
- If an intersection occurs outside the probability range 0 to 1, it is:
a) Always optimal
b) A saddle point
c) A pure strategy
d) Not a feasible mixed-strategy solution
Answer: d) Not a feasible mixed-strategy solution
- Before using the graphical method, one should usually:
a) Apply dominance to reduce the matrix
b) Add a constant to every game
c) Convert every probability to an integer
d) Remove all negative payoffs
Answer: a) Apply dominance to reduce the matrix
- A game larger than 2 × 2 may sometimes be reduced to 2 × 2 through:
a) Randomization
b) Dominance
c) Averaging
d) Maximax analysis
Answer: b) Dominance
- A general m × n zero-sum game can be formulated and solved using:
a) Queuing theory
b) Dynamic programming only
c) Linear programming
d) Inventory models
Answer: c) Linear programming
- In the linear-programming formulation, Player A seeks to:
a) Minimize every strategy probability
b) Eliminate constraints
c) Maximize the number of strategies
d) Maximize the guaranteed expected payoff
Answer: d) Maximize the guaranteed expected payoff
- In the linear-programming formulation, Player B seeks to:
a) Minimize the maximum expected payment to Player A
b) Maximize Player A’s payoff
c) Eliminate every column
d) Make all probabilities zero
Answer: a) Minimize the maximum expected payment to Player A
- The probabilities in a mixed-strategy linear program must:
a) Be unrestricted
b) Be nonnegative and sum to one
c) All be integers
d) All equal one
Answer: b) Be nonnegative and sum to one
- The linear programs for Players A and B are related through:
a) Simulation
b) Dominance only
c) Duality
d) Queuing balance
Answer: c) Duality
- The optimal objective values of the primal and dual game formulations are:
a) Always different
b) Opposite in sign only
c) Undefined
d) Equal under standard conditions
Answer: d) Equal under standard conditions
- If all payoffs are negative, a constant may be added to every entry to:
a) Simplify the linear-programming transformation
b) Change the optimal strategies
c) Eliminate the zero-sum property
d) Create dominance automatically
Answer: a) Simplify the linear-programming transformation
- After adding a constant (K) to all payoffs, the original game value is obtained by:
a) Adding (K) again
b) Subtracting (K) from the transformed value
c) Multiplying by (K)
d) Dividing by (K)
Answer: b) Subtracting (K) from the transformed value
- Adding the same constant to all matrix entries changes:
a) The optimal mixed strategies
b) The dominance order completely
c) The game value but not the optimal strategies
d) The number of players
Answer: c) The game value but not the optimal strategies
- The expected payoff of a mixed strategy is calculated using:
a) The largest payoff only
b) The smallest payoff only
c) The number of strategies
d) Probability-weighted payoffs
Answer: d) Probability-weighted payoffs
- The fundamental objective of an optimal strategy is to:
a) Guarantee the best possible outcome against an intelligent opponent
b) Maximize the number of available strategies
c) Ensure the opponent never receives a payoff
d) Select every strategy equally
Answer: a) Guarantee the best possible outcome against an intelligent opponent
- Which statement best summarizes two-person zero-sum game analysis?
a) Every game has a saddle point
b) Pure strategies apply when maximin equals minimax; otherwise mixed strategies may be required
c) Dominance always determines the final game value directly
d) Randomization eliminates the need to calculate expected payoffs
Answer: b) Pure strategies apply when maximin equals minimax; otherwise mixed strategies may be required
